However, Emmanuel Ojukwu, a police spokesman, could not confirm the death toll, saying only that the number of dead was in the dozens.
Ojukwu said the police had arrested at least 100 fighters after the clashes.
Mohammed Barau, another police spokesman, said that the fighters belonged to a group called Boko Haram, which is sometimes referred to as the Nigerian Taliban and wants to impose sharia – or Islamic law – on all of Nigeria.
Bauchi is one of 12 northern states in northern Nigeria where sharia is practiced. It lies along a line that roughly splits the country’s predominantly Muslim population in the north from the mainly Christian and animist communities in the south.
One of Boko Haram’s members, who was wounded during the clashes, was quoted as telling the Reuters news agency that his group wanted to “clean the [Nigerian] system which is polluted by western education and uphold sharia all over the country”.
“The police has been arresting our leaders that is why we decided to retaliate,” the man, who gave his name only as Abdullah, said.
The group was founded in 2004, setting up a base dubbed “Afghanistan” in the village of Kanamma village in the northern state of Yobe, close to the border with Niger.
Sectarian clashes between Muslims and Christians in Bauchi state in February left at least five people dead.
Muslims attacked Christians and set fire to churches in reatalitation for the burning of two mosques, which had been blamed on Christians.
Last November, more than 700 were killed in Jos, capital of Plateau state, when a political feud over a local election degenerated into bloody confrontation between the two religions.